Some Lie algebra structures on symmetric powers

TL;DR

The article generalizes an eigenvector-based construction of solvable Lie algebras on the symmetric powers $S^d(V^*)$ by using a linear map $\varphi:V\to V$ and an eigenvector $w$, showing the resulting bracket $[f,g]=g(w)\varphi_d^*(f)-f(w)\varphi_d^*(g)$ yields a Lie algebra that is solvable (nilpotent if $\varphi$ is nilpotent); it extends to an infinite-dimensional case on $S(V^*)$ and provides two concrete examples illustrating the method. It develops an isomorphism criterion via similarity transforms, reducing classification to orbits under $GL(V)$ actions, and specializes to the complex two-dimensional case where a complete list of isomorphism classes is obtained. The work also establishes a practical, parameter-counting perspective for classifying these algebras and demonstrates explicit infinite-dimensional realizations, highlighting connections to known low-dimensional Lie algebras and invariant-theoretic descriptions. Overall, it advances a structured, invariant-theory-based approach to constructing and classifying solvable Lie algebras on symmetric powers of dual spaces.

Abstract

Let $k$ be a field of any characteristic, $V$ a finite-dimensional vector space over $k$, and $S^d(V^*)$ be the $d$-th symmetric power of the dual space $V^*$. Given a linear map $\varphi$ on $V$ and an eigenvector $w$ of $\varphi$, we prove that the pair $(\varphi, w)$ can be used to construct a new Lie algebra structure on $S^d(V^*)$. We prove that this Lie algebra structure is solvable, and in particular, it is nilpotent if $\varphi$ is a nilpotent map. We also classify the Lie algebras for all possible pairs $(\varphi, w)$, when $k=\mathbb{C}$ and $V$ is two-dimensional.

Theorems & Definitions (32)

• Theorem 2.1
• proof
• Corollary 2.2
• Example 3.1
• Example 3.2
• Remark 3.3
• Theorem 4.1
• proof
• Corollary 4.2
• proof